Stress in formations from azimuthal variation in acoustic and other properties

ABSTRACT

The present disclosure is related to methods and apparatuses for acoustic velocity well logging. The method may include estimating a magnitude of a principal horizontal stress in a borehole in a formation. The method may include obtaining a far field stress orientation and making a measurement of near borehole stress orientation. The present disclosure also includes an apparatus configured to be conveyed into a borehole and perform the method. Formation stresses and directions may be estimated.

BACKGROUND OF THE DISCLOSURE

The present disclosure is related to the field of acoustic velocity welllogging. More specifically, the present disclosure is related to methodsof processing receiver signals from an acoustic well logging instrumentto determine certain shear wave propagation properties and stressproperties of earth formations.

BACKGROUND OF THE ART

In an anisotropic formation, shear waves travel at different velocitieswith different propagating directions and polarizations. In boreholeacoustic logging, the receivers are placed along the borehole axis sothat only the wave traveling along the borehole axis is measured.Borehole acoustic logging can measure the shear wave anisotropy withdifferent polarizations around the borehole which is sometimes referredto as the azimuthal anisotropy.

In most cases, an anisotropic rock can be modeled as a transverseisotropic (TI) material. For example, layered structures such as thestructure of shale or layered fractures inside a rock can cause suchanisotropy, which is sometimes referred to as intrinsic anisotropy. Thismaterial has one symmetry axis of infinite-fold rotational symmetry thatis perpendicular to the layers. When the rock's symmetry axis isparallel to the borehole axis, there will be no observable shear waveanisotropy from acoustic logging since the shear modes propagating alongthe axis for this geometry have the same velocity regardless of thedirection of polarization. This kind of configuration related to theborehole is sometimes referred to as vertically transverse isotropy(VTI).

If there is an angle between the symmetry axis and the borehole axis,the measured shear modes have two phase velocities corresponding to fastand slow modes with perpendicular polarization directions. In boreholedipole acoustic logging, azimuthal anisotropy may be observed whendipole modes are excited at different azimuthal directions. Theconfiguration in which the rock's symmetry axis is perpendicular to theborehole axis is sometimes referred to as horizontally transverseisotropy (HTI). For HTI, the shear mode that is polarized along thefracture (or layer) direction has a faster velocity than the shear modepolarized perpendicular to the fractures.

Azimuthal anisotropy may also be induced by stress in earth formations.Before a borehole is drilled, the rock itself may be pre-stressed.Stress can change the rock's elastic properties so that the shear wavepolarized along the largest principal stress may have a different shearvelocity than shear waves polarized perpendicular to the largestprincipal stress. This kind of anisotropy has a different character thanthe intrinsic anisotropy in borehole acoustic logging. The stress willredistribute around the borehole after the well is drilled, so that thestress distribution (both its magnitude and direction) near the boreholemay be very different from that far away from the borehole. The latteris considered to have the same stress condition as before the boreholeis drilled. This stress re-distribution may cause the shear velocity tovary in both azimuthal and radial directions. A formation with intrinsicanisotropy is homogeneous around the borehole area, but thestress-induced anisotropy in such a medium is non-uniform. The presentdisclosure is directed towards practical methods to distinguish betweenthese two kinds of azimuthal anisotropy caused by either fractures orstress, and to further characterize the stress-induced component.

SUMMARY OF THE DISCLOSURE

One embodiment of the present disclosure is a method for characterizingan earth formation comprising: making at least one measurement using alimited aperture source indicative of a principal direction of a nearfield stress in a borehole penetrating the earth formation; andobtaining one of: (i) an indication of a principal direction of a farfield stress in the earth formation and (ii) a magnitude of a horizontalprincipal stress; using the at least one measurement indicative of theprincipal direction of the near field stress and one of: (i) theindication of the principal direction of the far field stress in theearth formation and (ii) the magnitude of the horizontal principalstress for estimating the other one of: (i) the indication of theprincipal direction of the far field stress and (ii) the magnitude ofthe horizontal principal stress.

Another embodiment of the present disclosure is an apparatus forcharacterizing an earth formation comprising: a logging tool configuredto obtain at least one limited aperture measurement in a boreholepenetrating the earth formation; and at least one processor configuredto: (i) estimate a principal direction of a near field stress near theborehole using the at least one limited aperture measurement; and (ii)use the estimated principal direction of the near field stress and oneof: (i) an indication of a principal direction of a far field stress and(ii) a magnitude of a horizontal principal stress to estimate the otherone of: (i) the indication of the principal direction of the far fieldstress and (ii) the magnitude of the horizontal principal stress.

Another embodiment of the present disclosure is a non-transitorycomputer-readable medium product having stored thereon instructionsthat, when read by at least one processor, causes the at least oneprocessor to execute a method, the method comprising: using at least onemeasurement of claim by a limited aperture source indicative of aprincipal direction of near field stress in a borehole; and obtainingone of: (i) an indication of a principal direction of a far field stressin the earth formation and (ii) a magnitude of a horizontal principalstress; using the at least one measurement indicative of the near fieldstress direction and one of: (i) the indication of the far field stressdirection and (ii) the magnitude of the horizontal principal stress forestimating the other one of: (i) the indication of the principaldirection of the far field stress and (ii) the magnitude of thehorizontal principal stress.

BRIEF DESCRIPTION OF THE DRAWINGS

The file of this patent contains at least one drawing executed in color.Copies of this patent with color drawing(s) will be provided by thePatent and Trademark Office upon request and payment of the necessaryfee.

For detailed understanding of the present disclosure, reference shouldbe made to the following detailed description of an exemplaryembodiment, taken in conjunction with the accompanying drawing and inwhich:

FIG. 1 shows a schematic illustration of a wireline logging system;

FIGS. 2A-2C shows the radial, tangential and azimuthal shear stressesaround a borehole;

FIGS. 3A and 3B show plots of velocities of shear waves with x- andy-polarizations for the stress distribution of FIG. 2;

FIG. 4 illustrates the radial variation of the velocities of the fastand slow shear waves in a pre-stressed medium with a drilled borehole;

FIG. 5 shows a model used for a numerical simulation according to oneembodiment of the present disclosure;

FIG. 6 shows the fast and slow shear wave dispersion curves for anintrinsic HTI formation obtained using exemplary finite difference andfinite element simulations;

FIG. 7 shows fast and slow shear wave dispersion curves for a formationwith stress-induced anisotropy;

FIG. 8 shows an exemplary variation of one principal stress as afunction of azimuth and distance for an exemplary deviated borehole;

FIG. 9 a shows the distribution of velocity of a shear wave polarizedalong the x-direction in a deviated borehole for the stress distributionof FIG. 8;

FIG. 9 b shows the distribution of velocity of a shear wave polarizedalong the y-direction in a deviated borehole for the stress distributionof FIG. 8;

FIG. 10 shows constraints on the formation stress as a function ofazimuth for the stress distribution of FIG. 8;

FIG. 11 is a flow chart illustrating some steps of one embodimentaccording to the present disclosure;

FIG. 12 shows data and results of processing in a vertical borehole thatshows stress-induced anisotropy in sands and intrinsic anisotropy inshales; and

FIG. 13 shows data and results in a deviated borehole showingstress-induced anisotropy in sands and intrinsic anisotropy in shales.

DETAILED DESCRIPTION OF THE DISCLOSURE

The present disclosure is discussed with reference to specific logginginstruments that may form part of a string of several logginginstruments for conducting wireline logging operations. It is to beunderstood that the choice of the specific instruments discussed hereinis not to be construed as a limitation and that the method of thepresent disclosure may also be used with other logging instruments.

A well logging apparatus suitable for performing the limited apertureand dipole measurements disclosed herein is depicted in FIG. 1. Herein,a limited aperture measurement refers to an aperture that is narrowenough to define at least eight circumferential bins around a borehole.An acoustic array borehole logging tool, shown generally at 10, may beattached to one end of an armored electrical cable 8. The cable 8 may beextended into a borehole 2 penetrating earth formations, as showngenerally at 6A and 6B. A winch 18, or similar device known to thoseskilled in the art, may extend the cable 8 into the borehole 2. Theborehole 2 is typically filled with a liquid 4 which is known to thoseskilled in the art as “drilling mud”, or similar fluid. The liquid 4enables transmission of acoustic energy from the tool 10 outwardly tothe wall of the borehole 2.

Acoustic transmitters 12 are disposed on the tool 10 and shown generallyat 12. These transmitters 12 may include, but are not limited to, dipolesources, monopole sources, ultrasonic horns, and sources coupled to theborehole wall. The transmitters 12 may periodically emit acoustic energypulses 22. The pulses 22 typically travel radially outwardly from thetransmitter 12 through the fluid 4 in the borehole 2 until they strikethe wall of the borehole 2. The pulses 22 then typically travel alongthe borehole 4 wall. Some of the acoustic energy returns to the fluid 4in the borehole 2 and can be detected by a plurality of receivers 14that may be disposed axially on the tool. Typically, the receivers 14are spaced apart from the transmitter 12. In some embodiments, thereceivers 14 may include dipole receivers. In some embodiments, theremay be as few as one transmitter 12 and as few as one receiver 14.

The receivers 14 generate electrical signals corresponding to theamplitude of the acoustic energy reaching the receivers 14. In someembodiments the receivers 14 may include limited aperture receiversconfigured for logging-while-drilling (LWD). In other embodiments thetransmitters 12 may include separate high and low frequency transmitterswhere the high frequency transmitters may correspond to the dipolereceivers and the low frequency transmitters may correspond to thelimited aperture receivers. Alternatively, the high frequencytransmitters may correspond to the limited aperture receivers and thelow frequency transmitters may correspond to the dipole receivers. Instill other embodiments the transmitters 12 may only operate at lowfrequencies. It should further be noted that a limited aperturemeasurement may be obtained using a single monopole source in contactwith the borehole wall and a single receiver. Furthermore, a limitedaperture measurement may also be obtained using a single dipole sourceand a single dipole receiver.

An acoustic array borehole logging tool 10 typically includes signalprocessing electronics 16 which may digitize the signals from thereceivers 14 and impart the digitized signals to the cable 8. Signalsimparted to the cable 8 may be transmitted to a processor 20 at thesurface.

It is well known that stress can change the shear velocities of a rock,and thus may induce anisotropy in the formation. For example, if a shearwave is propagating in a direction perpendicular to the direction ofcompressive stress, then the velocity of the shear wave polarizedparallel to the direction of compressive stress may be larger than theshear wave polarized perpendicular to the direction of compressivestress. It follows then that if there are two perpendicular compressivestresses, σ_(x) and σ_(y), and the shear waves propagate along the zdirection, then the velocities of the two shear waves polarized alongthe x and y directions can be expressed as:

$\begin{matrix}\left\{ \begin{matrix}{v_{x}^{2} = {v_{0\; x}^{2} + {S_{//}\sigma_{x}} + {S_{\bot}\sigma_{y}}}} \\{{v_{y}^{2} = {v_{0\; y}^{2} + {S_{//}\sigma_{y}} + {S_{\bot}\sigma_{x}}}},,}\end{matrix} \right. & (1)\end{matrix}$

where v_(0x) and v_(0y) are the initial shear velocities polarized alongthe x and y directions without stress, and S_(//) and S_(∠) are thestress-velocity coefficients. If the two stresses, σ_(x) and σ_(y), aredifferent, then the velocities of the two shear waves are different aswell. Thus, the anisotropy is induced.

From laboratory measurements it has been observed that sandstonesusually have large stress-velocity coefficients. The values of S_(//)and S_(∠) of the sandstones with high porosities are higher than thoseof the sandstones with low porosities; however, the values of S_(//) andS_(∠) may be smaller in shale and may often be neglected.

Unlike the situation with intrinsic anisotropy, where the elasticityproperties of a rock are homogeneous around a borehole, anisotropyinduced by stress becomes a complicated issue when a borehole existssince the properties of the rock may no longer be homogeneous. Forexample, in a situation where an earth formation is under uniform stressbefore a borehole is drilled, after the borehole is drilled, the stresswill redistribute around the near-borehole area. At an infinitedistance, the stress may not change from its initial state. Assume thatthe borehole is drilled along the z direction through a linear elasticmedium. Also consider that there are two stresses at infinity, σ_(x) andσ_(y), and a fluid at a pressure p inside borehole. The resulting stressaround the borehole can be calculated analytically in cylindricalcoordinates (r, θ) by:

$\begin{matrix}\left\{ \begin{matrix}{\sigma_{r} = {{\frac{\sigma_{x} + \sigma_{y}}{2}\left( {1 + \frac{R^{2}}{r^{2}}} \right)} + {p\frac{R^{2}}{r^{2}}} + {\frac{\sigma_{x} - \sigma_{y}}{2}\left( {1 + {3\frac{R^{4}}{r^{4}}} - {4\frac{R^{2}}{r^{2}}}} \right)\cos\; 2\theta}}} \\{\sigma_{\theta} = {{\frac{\sigma_{x} + \sigma_{y}}{2}\left( {1 + \frac{R^{2}}{r^{2}}} \right)} - {p\frac{R^{2}}{r^{2}}} - {\frac{\sigma_{x} - \sigma_{y}}{2}\left( {1 + {3\frac{R^{4}}{r^{4}}}} \right)\cos\; 2\theta}}} \\{{\sigma_{r\;\theta} = {{- \frac{\sigma_{x} + \sigma_{y}}{2}}\left( {1 - {3\frac{R^{4}}{r^{4}}} + {2\frac{R^{2}}{r^{2}}}} \right)\cos\; 2\theta}},}\end{matrix} \right. & (2)\end{matrix}$

where θ is the angle between r and x, and R is the borehole radius.Similar equations characterize the stresses around a non-linear elasticmedium.

FIG. 2A shows the radial stress (σ_(rr)) around an exemplary boreholehaving a diameter of 8.5 inches (21.6 cm) when the initial stressesbefore the borehole was drilled were σ_(x)=−50 MPa (i.e., under tension)and σ_(y)=30 MPa. (Note: these values are for exemplary purposes only toillustrate the effects that may occur.) FIG. 2B shows the tangentialstress (σ_(θθ)) and FIG. 2C shows the radial-azimuthal shear stress(σ_(rθ)). These figures show that the stress around the borehole variesin both azimuthal and radial directions.

The complex distribution of stress around a borehole may cause variationof the velocities of shear waves with different polarizations. Consideran example where stress-velocity coefficients in eqn. (1) are:S_(//)=89213 (m/s)²/MPa and S_(∠)=31867 (m/s)²/MPa, which are typical ofa sandstone, and the stress condition and borehole size are the same asthe previous example. The distribution of the resulting velocities ofvertically propagating shear waves with x- and y-polarizations areplotted in FIGS. 3A, and 3B respectively. Red areas have higher shearvelocities than blue areas. These two figures have the samevelocity-color mapping so that the velocities may be compared by thecolors. As shown in FIG. 4, along the x direction the velocity of thex-polarized shear wave is larger at infinity than that near the borehole403; however, along the y direction the velocity of the y-polarizedshear wave is smaller at infinity than that near the borehole 401. Thetwo curves of the two shear velocities cross over each other at acertain radial distance.

On the other hand, for a formation having intrinsic anisotropy, sincethe formation is homogenous there is no difference between the far-fieldor near-field velocities. The shear velocity should be constant for eachwave with a particular polarization.

One way to distinguish these two kinds of azimuthal anisotropy is toidentify if there is a difference between the far-field and near-fieldvelocities and if the velocities vary azimuthally as well. In boreholeacoustic logging, this may be achieved by using broad band frequencysources and receivers. The energy of a low frequency dipole mode canreach deep into an earth formation to obtain information from far awayfrom the borehole. In contrast, the high frequency dipole mode may onlybe used to obtain information about the earth formation near theborehole.

In some embodiments, borehole acoustic logging may be achieved by usinga low frequency cross dipole mode to obtain information that is distantfrom the borehole and limited aperture log data to explore the formationnear the borehole. The limited aperture log data may include one or morenear field measurements. In other embodiments, the limited aperture logdata may be used to explore the formation near the borehole and becombined with prior acoustic logging information that characterizes theformation far from the borehole. Since the limited aperture logging maybe performed while drilling, some embodiments enable the boreholeacoustic logging to be performed in real-time. By using a limitedaperture tool having a narrow aperture at high frequencies, it ispossible to measure the compressional wave velocity in the earthformation as a function of azimuth. In contrast to conventional acoustictools that operate in the frequency range of 2-10 kHz, a limitedaperture tool may have a narrow aperture and can make high frequencymeasurements at frequencies such as 20-100 kHz.

The exemplary borehole used for all of the models for numericalsimulation is shown in FIG. 5 and has a diameter of 8.5 inches witheither a fast or a slow formation around it. In a fast formation, theshear velocity is greater than the compressional wave velocity in theborehole fluid so that a recognizable shear wave can be detected by alogging tool. The borehole fluid is 13.5 PPG oil based mud, whosedensity is 1.62 g/cc and velocity is 1246 m/s (or slowness is 245μs/ft). The positions of the acoustic sources and receivers in the modelsimulate those of an XMAC Elite® cross-dipole acoustic logging tool ofBaker Hughes Incorporated.

Both layered media (such as shale) and formations with orientedfractures can be modeled by the intrinsic anisotropy models. Most of theanisotropic earth formations are considered to be transversely isotropic(TI) media. They have five independent elasticity constants. In orderfor the acoustic dipole logging to detect two distinct shear velocities,there must be an angle between the symmetry axis of the TI medium andthe borehole axis. For purposes of simplicity, only the HTIconfiguration is considered because the two axes are perpendicular toeach other. The elastic tensor of these media may be represented in thefollowing form,

$\begin{matrix}{{C = \begin{bmatrix}c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\c_{12} & c_{22} & c_{12} & 0 & 0 & 0 \\c_{13} & c_{12} & c_{11} & 0 & 0 & 0 \\0 & 0 & 0 & c_{44} & 0 & 0 \\0 & 0 & 0 & 0 & c_{55} & 0 \\0 & 0 & 0 & 0 & 0 & c_{44}\end{bmatrix}},,} & (3)\end{matrix}$where c₁₃=c₁₁−2 c₅₅. The layers or fractures are parallel to x-z plane.Considering the shear waves propagating along the z axis (boreholeaxis), the speeds of shear waves polarized along x and y directions are

$\begin{matrix}{{v_{zx} = \sqrt{\frac{c_{55}}{\rho}}}{{v_{zy} = \sqrt{\frac{c_{44}}{\rho}}},,}} & (4)\end{matrix}$where ρ is the density of the formation. Usually, c₅₅>c₄₄ andv_(xz)>v_(zy).

Density Elastic Constants (GPa) (g/cc) C₁₁ C₂₂ C₁₂ C₁₃ C₄₄ C₅₅ Formation2.35 22.59 15.07 5.1 8.03 5.26 7.28The equivalent velocity and slowness of these two models are:

P-wave Shear P-wave Shear velocity velocity slowness slowness (m/s)(m/s) (μs/ft) (μs/ft) Fast Slow Fast Slow Fast Slow Fast Slow Formation3100 2532 1760 1496 98.3 120.4 173.2 203.7

A stress-induced anisotropy model may now be defined. To configure thisstress-induced anisotropy model, an isotropic formation may be modeledusing its two stress-velocity coefficients, S_(//) and S_(∠).

Shear X direction P-wave wave S_(//) S⊥ horizontal Density velocityvelocity (m/s)²/ (m/s)²/ compression (g/cc) (m/s) (m/s) MPa MPa (MPa)Originally 2.2 2900 1580 89213 31867 15 isotropic formation

FIG. 6 shows an example of dipole modes for formations with intrinsicanisotropy. The 3D finite difference (FD) and finite element (FE)results for both the fast and slow dipole modes are shown in FIG. 6 foran HTI formation. In the above referenced model configuration, the fastdipole mode has its direction along the x axis. In the case of intrinsicanisotropy, the fast dipole mode remains as the fast mode polarizedalong the x direction in the entire frequency range, while the slowdipole stays as the slow mode with polarization along the y direction.The open circles, 601 & 603, are for FD modeling and the curves, 605 &607, are for FE modeling.

FIG. 7 shows an example of stress-induced anisotropy in an earthformation. The 3D-FD, and 3D-FE results of both the fast and slow dipolemodes are shown in FIG. 7. At low frequency, the XX dipole mode, 703 &707, (polarized along x direction) reaches the fast shear velocitycaused by the compression along the x direction, and the YY dipole mode(polarized along y direction) reaches the slow shear velocity, 701 &705. At frequencies higher than 4 kHz, the XX dipole becomes the slowmode while the YY dipole becomes the fast mode. Therefore, the fast modeat high frequency is polarized along the y direction which is 90°different from the polarization direction (the x direction) of the fastmode at low frequency. The phenomenon of azimuthal polarization anglechange of the fast dipole mode at low and high frequencies is unique tostress-induced anisotropy and can be used to distinguish it fromintrinsic anisotropy.

Based on the above theory, one embodiment of the present disclosureprocesses the azimuthal angles of the fast dipole modes at both low andhigh frequencies and calculates the angle difference to identify theintrinsic or stress-induced anisotropy. This method may use an azimuthalanisotropy analysis program for borehole acoustics that may beconfigured to calculate the azimuthal polarization angles of fast andslow dipole waves, as well as the anisotropy magnitude.

The azimuthal anisotropy analysis program may be first applied to getthe azimuthal polarization angle α_(L) of the fast dipole mode at lowfrequency (for example, between 0.5 kHz to 3 kHz). Next, the waveformsare filtered to keep the wave components of higher frequencies (forexample, 4.5 kHz and higher) and the azimuthal anisotropy analysisprogram may be used to process them again. This time the azimuthalpolarization angle of the fast dipole mode at high frequency is α_(H).According to the subject theory, if the anisotropy is caused by stress,then the difference between α_(L) and α_(H) is 90 degrees. If theazimuthal angle of the two fast dipole modes are almost the same (thatis, the difference is close to 0 degrees), then the anisotropy isintrinsic or caused by fractures. Sometimes the angle difference mightbe away from both 0 and 90 degrees (for example, around 45 degrees). Inthis situation the dispersion curve cross-over method will have adifficulty in detecting the cross-over. Note that this disclosure doesnot need to calculate the fast and slow waveforms and does not need tocalculate the dispersion curves.

The cross-over method requires the use of the two principal directionsof the dipole waves (the azimuthal angle of fast and slow dipole waves)at low frequencies to separate the original waveforms into the fast andslow waves that contain high frequency components. In contrast, it isassumed herein that the two principal directions of low frequency wavesare also the principal directions of the high frequency waves.

There is an implicit assumption in the foregoing that the orientationsof the fast and slow dipole modes in the well as measured by thefrequency-dependent dipole mode propagation properties do not varyslowly with distance from the wellbore wall, but rather that they may“flip” with the fast direction in the far-field becoming the slowdirection close to the well. In fact, this “flip” in relative guidedmode velocity with frequency is used as a diagnostic discriminator todifferentiate between stress-induced anisotropy and anisotropy inducedby earth layering. The aspect of the present disclosure which isdiscussed next does not make this assumption.

In one embodiment of the disclosure, an assumption may be made that thevertical stress is a principal stress. It is known in the art toestimate the magnitude of the vertical stress by computing the weight ofoverlying rocks and fluids, and it is possible to measure the minimumprincipal stress from hydraulic fracturing or from extended leakofftests. The orientation of the maximum horizontal stress can sometimes bedetermined from characteristics of wellbore failures. When wellborefailures are detected, one can often constrain the stress magnitudesbecause the width of a breakout or the occurrence of a drilling-inducedtensile fracture can be directly related to the stresses acting aroundthe well, and model representations of these features can be expressedin a manner similar to an image log or as a cross-section cutperpendicular to the wellbore. By matching such a model to such anobservation, it is possible to constrain stress magnitudes andorientations. In order to use such constraints on the magnitude of themaximum horizontal stress, the rock strength that resists failure shouldbe known a priori; this is often difficult to determine, as is known tothose of ordinary skill in the art.

Using the assumption that the vertical stress is a principal stress, thestress distribution around a deviated borehole can be modeled usingmethods known to one of skill in the art. The distribution of stressesaround a deviated borehole is much more complex than the distribution ofstresses around a vertical borehole discussed previously.

Shown in FIG. 8 is the maximum principal stress S1 (ordinate) as afunction of azimuth (abscissa) around an exemplary deviated borehole atdistances equal to 1.01R (801), 1.1R (803), 1.2R (805), 1.3R (807), 1.4R(809) and 1.5R (811), where R radius of the borehole. Also shown in FIG.8 is a curve 823 showing the azimuth as a function of distance from thewell where the maximum principal stress is greatest for each radialdistance. The difference in azimuth between the point 821 (which is onthe borehole wall) of curve 823 and the point 825 of curve 823 (which isdistant from the borehole wall) may not be 90 degrees.

Using such a modeled stress distribution, it is possible to predict thevelocities of elastic waves in the earth formation. FIG. 9 a shows thedistribution of shear wave velocities around the borehole for shearwaves polarized in the x-direction corresponding to the exemplary stressdistribution of FIG. 8. FIG. 9 b shows the distribution of shear wavevelocities around the borehole for shear waves polarized in they-direction corresponding to the exemplary stress distribution of FIG.8.

Referring again to FIGS. 9 a-9 b, it can be seen that the x-polarizedshear wave at location 903 is faster than the y-polarized shear wave atlocation 903′. It can also be seen that x-polarized shear wave atlocation 901 is slower than the y-polarized shear wave at location 901′.Thus, a crossover as predicted by the simple model does occur. Incontrast, FIGS. 9 a and 9 b also show that the azimuth angle at theborehole wall corresponding to the fastest and slowest velocities is notthe same as the azimuth angle at the far offset, and the difference isnot 90 degrees. The azimuthal P-wave velocity measurements made by ahigh frequency limited aperture tool discussed above allow for thedetermination of the orientation of the stress field near the borehole.The far field stress orientation may be estimated from the cross-dipolemeasurements or from prior measurements in other wells nearby. Insteadof using a compressional wave velocity measurement, in a fast formationa direct measurement may be made of the shear velocity. In addition tothe difference between the two directions not being 90°, there may be agradual transition of the velocities from the borehole wall to the fardistances thereby showing that there may not be a single crossover angleat which the slow and fast modes are interchanged.

In one embodiment of the disclosure, analysis 1100 proceeds according tothe flow chart shown in FIG. 11. In step 1101, limited aperture log datamay be acquired over a range of depths in the borehole. The limitedaperture log data may be acquired during drilling. The limited aperturelog data may be transmitted to a processor located at the surface ordownhole. In step 1102, cross-dipole data may be acquired over a rangeof depths in the borehole. In some embodiments, the cross dipole datamay be acquired at the same time as the limited aperture log data ofstep 1101. In step 1103, the orientations of fast and slow dipole modesin a wellbore as a function of frequency may be determined. This stepmay include a generalization of the methods described in Alford, 1986.One approach to generalizing this method, which is not intended to berestrictive, includes: using a band pass filter to filter the waveformarrivals at a series of receivers over a series of narrow frequencybands; independently rotate each set of the band passed data into theprincipal planes within the time range of each signal in which thebending mode arrives; and measure the angles relative to the borehole inwhich the fast and slow waves “bend” the wellbore within each frequencyband using optimization criteria based on the ability to distinguish twounique signals within each frequency band. In step 1105, at least oneproperty related to the orientation of the far-field maximum stress fromthe orientation of the fast direction at low frequency may bedetermined. One property of the horizontal maximum stress orientationmay be the orientation itself. The at least one property may include adirection that is contained in a plane containing the orientation.Finally, in step 1107, constraints may be estimated.

In some embodiments, estimating constraints may include using knowledgeof the wellbore orientation and/or relationships between far-fieldstress magnitudes and orientations and the orientations and magnitudesof stresses near a wellbore. In other embodiments, constraints may beestimated for relationships between two or more properties including,but not limited to, the following properties:

-   a. The far-field maximum principal stress magnitude-   b. The far-field minimum principal stress magnitude-   c. The far-field intermediate principal stress magnitude-   d. Three angles that define the principal stress orientations, one    description of which is:    -   i. The inclination of the far-field maximum stress    -   ii. The azimuth of the projection of the far-field maximum        stress into a horizontal plane    -   iii. The rake of the intermediate stress, S2 (that is, the angle        from the horizontal to the intermediate stress in the S1-S2        stress plane).        In some embodiments, step 1107 may utilize the same mathematical        methods as are described in Peska and Zoback, 1995. These        methods and their application are discussed in Moos (2007) and        Zoback et al. (2005). In one implementation that is not intended        to be restrictive, plots such as are found in the Users' Manual        of GMI•SFIB can be used to define relationships among unknown        parameters. Other relationships between stresses in the far        field and near the wellbore may be utilized, including        relationships that relate stresses to physical properties of the        rock or to changes in those physical properties

A plot of the orientation of the fast dipole as a function of frequencywithin the ranges of each set of band-pass filtered data may reveal theamount of rotation. The angle at the highest frequency band may indicatethe orientation near the wellbore wall and the angle at the lowestfrequency band may be related to the orientation of the far fieldstress. If the difference is near 0 degrees, it is likely that theanisotropy is due to intrinsic properties of the formation and is notstress-induced. If the difference is near 90 degrees, then thedifference may indicate that the anisotropy is stress-induced and thatthe well is likely drilled in a principal stress plane. Finally, if thedifference is less than 90 degrees and more than 0 degrees, its valuecan be used to estimate properties of the stress field. The angles inthe different frequency bands can be estimated by band-passing thecross-dipole data and performing a coordinate rotation. Details of thecoordinate rotation are well known in the art. See, for example, Alford(1986). In one embodiment, the values of the velocities of the fastshear mode and the slow shear mode may be used to estimate the stresses.Eqn. (1) may be used for the purpose.

In another embodiment of the disclosure, the azimuth corresponding tothe maximum stress at the borehole may be used to estimate upper andlower bounds on the stresses. Peska and Zoback disclose how to determinea stress magnitude from the rotation of a breakout azimuth away from theazimuth of the far-field maximum horizontal stress. Knowledge of thisrotation allows one or more unknown properties of the stress field to bederived using a priori knowledge of other properties of the stressfield. In the discussion that follows, it is assumed that the knownproperties are the magnitude of S_(v), and the magnitude of S_(hmin).

FIG. 10 is plotted assuming that the vertical stress is a principalstress, for simplicity and illustration. In FIG. 10, the abscissa is theazimuth of the far-field maximum horizontal stress and the ordinate isthe magnitude of the maximum far-field principal stress in pounds pergallon equivalent density (PPG). The colored band 1001 on this plotcorresponds to the only combination of azimuth and magnitude for which abreakout in a given orientation (in this case the orientation of 821 inFIG. 8), could form in a particular well with a known deviation anddeviation azimuth and for a known overburden and least horizontalprincipal stress magnitude. The width of the band may be due touncertainty in the azimuth of the breakout. Such uncertainty may be aknown property of observations of breakouts in wells. One edge of theband may correspond to the stress parameters required to match one limitof the range of breakout azimuths and the other edge of the band maycorrespond to the stress parameters required to match the other limit ofthe range of breakout azimuths. In using the information contained inFIG. 10, if the azimuth of the far-field stress is uncertain, then twovertical lines corresponding to the upper and lower limits of theazimuths of the far-field stress can be used and the range of possiblestress magnitudes may lie between the upper and lower bounds of the fourpoints on the edges of the colored band 1001 corresponding to theintersections of each of these vertical lines.

The vertical stress can be found by integrating the density of overlyingrock; the minimum principal stress magnitude can be found methods knownto those of skill in the art, such as fracture closure pressure, leakofftests, and Eaton's or other known methods for pore pressure/fracturegradient prediction. The wellbore orientation can be measured in avariety of ways, such as by conventional survey techniques.

The breakout orientation is the orientation at the wellbore where S1 isgreatest (821). That direction is also the direction corresponding tothe orientation around the well in FIG. 9 where the shear velocities maybe greatest (901). The near-well orientation where the shear velocitiesare greatest may be the orientation where the near-well stress isgreatest, which is typically where a breakout would form.

Because the breakout azimuth may be the same as the azimuth where theshear velocities are greatest (they are both controlled by the azimuthwhere S1 is greatest at the well) it is possible to replace the breakoutazimuth as an input to analyses such as in FIG. 10 with the azimuth ofthe high-frequency fast dipole (901). i.e., the position of the coloredband in FIG. 10 would be the same if the input were the azimuth of thehigh-frequency fast dipole or the high frequency limited aperturemeasurement of compressional or shear velocity instead of the azimuth ofa breakout. This is an advantage over using a breakout azimuth todetermine the azimuth near the well where S1 is greatest, because (1)breakouts do not always form, and (2) even if breakouts do form, it isnot always possible to measure their azimuth.

Using the information derived from the dipole analysis (the azimuths ofthe low-frequency fast dipole and of the high-frequency fast dipole) ordipole-limited aperture analysis (the azimuths of the low frequency fastdipole and of the high frequency limited aperture), it may be possibleto determine the maximum stress magnitude using the information in FIG.10. In some embodiments, the azimuths of the low frequency fast dipolemay be supplemented or replaced with azimuths obtained using, but notlimited to, seismic measurements, hydraulic fracture data, and imagelogs. This may be done by first determining the far-field maximumhorizontal stress azimuth from the azimuth of the fast dipole mode atlow frequency. In FIG. 10, the azimuth of the fast dipole mode at lowfrequency (far field) is represented by the vertical line 1003. Then thebounds on the stress are given by the values 1005 & 1007.

In the same manner as described above for analyses using the breakoutazimuth, the range of possible stress magnitudes may be due to the factthat the region of possible stress states has a finite width, the widthhaving been computed from an uncertainty in the observed azimuth of thebreakout or in the azimuth of the fast dipole at high frequency. Pairsof lines corresponding to the upper and lower limits of the range oforientations of the fast dipole mode at low frequency can be used todefine the range of possible stress magnitudes that also account foruncertainty in the far-field stress orientation. Stress orientations maybe obtained separately from stress magnitudes.

The example illustrated in FIGS. 9 a-9 b shows that given a stressdistribution, it is possible to predict the distribution of shearvelocities by azimuth, distance, and polarization. In one embodiment ofthe disclosure, a table lookup or other type of inversion is used toestimate the unknown parameters of the stress distribution frommeasurements of shear velocities. It is possible to use the differencebetween the near borehole (high frequency) and the far-field (lowfrequency) orientations to estimate the minimum and maximum horizontalstresses in the formation. The velocity measurements made atintermediate frequencies provide indications of the velocity atintermediate distances.

FIG. 12 shows an example of well logs acquired and processed withcross-dipole measurements. Track 1 1201 includes the gamma ray 1219 logwhich is an indication of shaliness. The caliper 1213 shows that theborehole has uniform gauge (12.5 inches). The tool azimuth 1215 isconstant, showing that the tool is not rotating. The borehole deviation1217 is close to 0 degrees. Track 3 1205 shows the slowness of the twoshear waves. Track 4 1207 shows the estimated anisotropy at theborehole. Track 5 1209 shows the azimuths of the fast and slow shearwaves at the borehole. Track 6 1211 is a plot of the difference betweenthe azimuthal polarization angles α_(L) and α_(H) defined above. Valuesof the difference in angles 1221 near 0 degrees, approximately 3 degreesor less, are suggestive of intrinsic anisotropy while greater values areindicative of the presence of stress induced anisotropy; and there maystill be intrinsic anisotropy. Values close to 90 degrees are indicativeof stress-induced anisotropy of a wellbore drilled in a principal stressplane. Note that in the shale interval 1251 intrinsic anisotropy isindicated while in the sand 1221 below (see gamma ray log 1219), stressinduced anisotropy is indicated.

FIG. 13 shows plots similar to FIG. 12 in a deviated well. Track 1 1301includes the gamma ray, caliper, azimuth, and borehole deviation. Track3 1305 shows the slowness of the two shear waves. Track 4 1307 shows theestimated anisotropy at the borehole. Track 5 1309 shows the azimuths ofthe fast and slow shear waves. Track 6 1311 is a plot of the differencebetween the azimuthal polarization angles α_(L) and α_(H). Note that inthe upper, shaly, interval there is intrinsic anisotropy while in thelower, sandy interval stress-induced anisotropy is indicated. Notefurther that in the upper portion of the shale interval 1321 thedifference is close to 20 degrees. This suggests that the shale intervalalso has some stress-induced anisotropy. An explanation of thedifference close to 0° in the vertical shale interval 1251 in FIG. 12 isthat the intrinsic anisotropy is much greater than the stress inducedanisotropy.

Those versed in the art and having benefit of the present disclosurewould recognize that the stress field in the subsurface is characterizedby five quantities. These quantities are the three principal stressesand two angles. The present method works because the vertical principalstress can be estimated from integration of the density log. Formeasurements in a vertical borehole, one of the angles is known. Thedifference between the near field and far field stress directionsprovides a constraint using FIG. 10 to estimate the horizontal principalstresses. In some embodiments, near field stress direction and themagnitude of the horizontal principal stress may be used to estimate thedirection of the far field stress.

Once the orientation and/or magnitudes of the principal stressdirections have been determined, this information can be used to controlfurther drilling operations. For example, U.S. Pat. No. 7,181,380 toDusterhoft et al, having the same assignee as the present disclosure andthe contents of which are fully incorporated herein by reference,teaches a process to determine optimal completion type and design priorto drilling of a hydrocarbon producing well. Moos (2006) disclosesmethods for selecting mud and casing design. Castillo et al (1987)disclose determination of maximum column height for assessment of theeconomic value of oil in place. U.S. Pat. No. 7,349,807 to Moos teachesthe evaluation of risk in pore pressure prediction and its content isalso fully incorporated herein by reference.

The present disclosure has been described above in terms of a wirelineimplementation, however, this is exemplary and illustrative only as themethod of the present disclosure may also be used in otherimplementations such as in a measurement-while-drilling (MWD)implementation.

The processing of the measurements made in wireline applications may bedone by the surface processor 20, by a downhole processor, or at aremote location. The data acquisition may be controlled at least in partby the downhole electronics. Implicit in the control and processing ofthe data is the use of a computer program on a suitable non-transitorymachine readable-medium that enables the processors to perform thecontrol and processing. The non-transitory machine-readable medium mayinclude ROMs, EPROMs, EEPROMs, flash memories and optical disks. Theterm processor is intended to include devices such as a fieldprogrammable gate array (FPGA).

While the foregoing disclosure is directed to specific embodiments ofthe present disclosure, various modifications will be apparent to thoseskilled in the art. It is intended that all variations within the scopeof the appended claims be embraced by the foregoing disclosure.

What is claimed is:
 1. A method for characterizing an earth formation,comprising: obtaining at least one limited aperture measurement atfrequencies greater than 20 kHz in a borehole penetrating the earthformation and obtaining a principal direction of a near field stressusing the at least one limited aperture measurement; and obtaining amagnitude of a horizontal principal stress; using the principaldirection of the near field stress and the magnitude of the horizontalprincipal stress for estimating a principal direction of a far fieldstress.
 2. The method of claim 1 wherein the at least one limitedaperture measurement further comprises a series of measurements over arange of frequencies over a range of depths of the borehole penetratingthe earth formation.
 3. The method of claim 1 further comprising makingthe at least one limited aperture measurement during drillingoperations.
 4. The method of claim 1 wherein the at least one limitedaperture measurement further comprises at least one of: (i) acompressional wave velocity and (ii) a shear wave velocity.
 5. Themethod of claim 1 further comprising: marking a selected depth of theformation as having intrinsic anisotropy if, at the selected depth, themagnitude of the difference between the principal direction of the nearfield stress and the principal direction of the far field stress is lessthan about 3 degrees.
 6. The method of claim 1 further comprising:marking a selected depth of the borehole as being in a principal stressplane of the earth formation if, at the selected depth, the magnitude ofthe difference between the principal direction of the near field stressand a normal to the principal direction of the far field stress is lessthan about 3 degrees.
 7. The method of claim 1 further comprisingestimating a magnitude of a principal stress of the formation using arelation between a velocity of a fast shear mode, a velocity of a slowshear mode, and a relation between the velocity of the fast shear mode,the velocity of the slow shear mode, and two principal stresses in theformation.
 8. The method of claim 1 further comprising conducting afurther operation selected from: (i) determining a completion type, (ii)designing a completion type, (iii) selecting mud for drillingoperations, (iv) designing a casing for completion, (v) determination ofeconomic value of a reservoir, and (vi) evaluation of a risk in porepressure prediction.
 9. The method of claim 1 further comprising:estimating a vertical stress at the borehole.
 10. An apparatus forcharacterizing an earth formation comprising: a logging tool configuredto obtain at least one limited aperture measurement at frequenciesgreater than 20 kHz in a borehole penetrating the earth formation; andat least one processor configured to: (i) estimate a principal directionof a near field stress near the borehole using the at least one limitedaperture measurement; (ii) obtain a magnitude of a horizontal principalstress; and (ii) use the estimated principal direction of the near fieldstress and a magnitude of a horizontal principal stress to estimate aprincipal direction of a far field stress.
 11. The apparatus of claim 10wherein the at least one limited aperture measurement further comprisesat least one of: (i) a compressional wave velocity and (ii) a shear wavevelocity.
 12. The apparatus of claim 10 wherein the at least oneprocessor is further configured to: mark a selected depth of the earthformation as having intrinsic anisotropy if, at the selected depth, amagnitude of a difference between the principal direction of the nearfield stress and the principal direction of the far field stress is lessthan about 3 degrees.
 13. The apparatus of claim 10 wherein the at leastone processor is further configured to mark a selected depth of theborehole as being in a principal stress plane of the formation if, atthe selected depth, a magnitude of a difference between the principaldirection of the near field stress and a normal to the principaldirection of the far field stress is less than about 3 degrees.
 14. Theapparatus of claim 10 wherein the at least one processor is furtherconfigured to a magnitude of a principal stress of the formation using arelation between a velocity of the fast shear mode, a velocity of a slowshear mode, and a relation between the velocity of the fast shear mode,the velocity of the slow shear mode, and two principal stresses in theformation.
 15. The apparatus of claim 10 wherein the at least oneprocessor is further configured to use conduct a further operationselected from: (i) determining a completion type, (ii) designing acompletion type, (iii) selecting mud for drilling operations, (iv)designing a casing for completion, (v) determination of economic valueof a reservoir, and (vi) evaluation of a risk in pore pressureprediction.
 16. The apparatus of claim 10 further comprising: aconveyance device configured to convey the logging tool into theborehole, the conveyance device selected from: (i) a wireline, and (ii)a drilling tubular.
 17. The apparatus of claim 10 wherein at least oneprocessor is further configured to determine a magnitude of a principalstress of the formation using the principal direction of the near fieldstress, the principal direction of the far field stress, and a verticalstress at the borehole.
 18. A method for characterizing an earthformation, comprising: obtaining at least one limited aperturemeasurement at frequencies greater than 20 kHz and obtaining a principaldirection of a near field stress in a borehole penetrating the earthformation using said limited aperture measurement; and obtaining anindication of a principal direction of a far field stress in the earthformation; using the principal direction of the near field stress andthe indication of the principal direction of the far field stress in theearth for estimating a magnitude of the horizontal principal stressusing a model in which there is a gradual transition of an azimuthaldirection of a fast shear mode from near the borehole to a far field.19. The method of claim 18 wherein the indication of the principaldirection of the far field stress is obtained using cross dipole logdata.
 20. The method of claim 18 further comprising estimating a furtherprincipal stress direction.
 21. The method of claim 18 furthercomprising estimating a magnitude of a principal stress of the formationusing a relation between a velocity of a fast shear mode, a velocity ofa slow shear mode, and a relation between the velocity of the fast shearmode, the velocity of the slow shear mode, and two principal stresses inthe formation.
 22. The method of claim 18 further comprising conductinga further operation selected from: (i) determining a completion type,(ii) designing a completion type, (iii) selecting mud for drillingoperations, (iv) designing a casing for completion, (v) determination ofeconomic value of a reservoir, and (vi) evaluation of a risk in porepressure prediction.